Convergence of SMACOF

TL;DR

This paper introduces mSMACOF, a modification of SMACOF that ensures proper convergence by rotating configurations to principal components, matching SMACOF's convergence speed while improving solution convergence properties.

Contribution

The paper proposes mSMACOF, a novel modification that guarantees convergence of SMACOF solutions through principal component rotation, without altering the stress values.

Findings

mSMACOF produces a convergent sequence of configurations.

Both SMACOF and mSMACOF have the same linear convergence rate.

The convergence speed is determined by the largest eigenvalue of the Guttman transform derivative.

Abstract

To study convergence of SMACOF we introduce a modification mSMACOF that rotates the configurations from each of the SMACOF iterations to principal components. This modification, called mSMACOF, has the same stress values as SMACOF in each iteration, but unlike SMACOF it produces a sequence of configurations that properly converges to a solution. We show that the modified algorithm can be implemented by iterating ordinary SMACOF to convergence, and then rotating the SMACOF solution to principal components. The speed of linear convergence of SMACOF and mSMACOF is the same, and is equal to the largest eigenvalue of the derivative of the Guttman transform, ignoring the trivial unit eigenvalues that result from rotational indeterminacy.